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November  2022, 18(6): 4151-4182. doi: 10.3934/jimo.2021152

Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit

Yu-Chung Tsao 1,2,3,4,, , Hanifa-Astofa Fauziah 1,5, , Thuy-Linh Vu 1,2, and  Nur Aini Masruroh 1,5,

1. 

Department of Industrial Management, National Taiwan University of Science and Technology, Taipei, Taiwan
2. 

Artificial Intelligence for Operations Management Research Center, National Taiwan University of Science and Technology, Taipei, Taiwan
3. 

Department of Business Administration, Asia University, Taichung, Taiwan
4. 

Department of Medical Research, China Medical University Hospital, China Medical University, Taichung, Taiwan
5. 

Department of Mechanical and Industrial Engineering, Universitas Gadjah Mada, Yogyakarta, Indonesia

* Corresponding author: Yu-Chung Tsao

Received  March 2021 Revised  June 2021 Published  November 2022 Early access  September 2021

In the modern global economy, trade credit financing is typical in business transactions for both sellers and buyers. The seller offers a credit period to attract new buyers or stimulate demand, and the buyer takes the opportunity to accumulate revenue. To obtain this benefit, the seller prefers trade credit policies that are dependent on the quantity ordered, referred to as order-linked trade credit. The buyer can obtain the benefits from a fully delayed payment if their order is sufficiently large. Similarly, the seller can sell many products while granting a credit period. Otherwise, the buyer receives only partial trade credit, and the seller can take the opportunity of both cash and credit payments. In this study, an economic order quantity (EOQ) inventory model for deteriorating products, under default risk control-based trade credit, is formulated using a discounted cash flow approach. The seller offers to the buyer order-linked trade credit with price-and credit-period-dependent demand. The optimal selling price, credit period policies, and replenishment cycle time are determined simultaneously, while maximizing the present value of the seller's total profit. Moreover, this research provides numerical examples and sensitivity analysis to illustrate the theoretical results, solution procedure, and gain managerial insights. 200 words.

Keywords: EOQ inventory model, deterioration product, order-linked trade credi, discounted cash-flow.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Yu-Chung Tsao, Hanifa-Astofa Fauziah, Thuy-Linh Vu, Nur Aini Masruroh. Optimal pricing, ordering, and credit period policies for deteriorating products under order-linked trade credit. Journal of Industrial and Management Optimization, 2022, 18 (6) : 4151-4182. doi: 10.3934/jimo.2021152
References:
[1]

A. A. A. Abuhommous, The impact of offering trade credit on firms' profitability, J. Cor. Account. Fina., 28 (2017), 29-40.  doi: 10.1002/jcaf.22298.

[2]

A. A. A. Aggarwal and S. P. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, J. Opers. Res. Soci., 46 (1995), 658-662. 

[3]

D. Atnafu and A. Balda, He impact of inventory management practice on firms' competitiveness and organizational performance: Empirical evidence from micro and small enterprises in ethiopia, Cogent. Busi. Mag., 5 (2018).

[4]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), 33-74.  doi: 10.2307/2152750.

[5]

A. Cambini and L. Martein, Generalized convexity and optimization: Theory and application, Springer-Verlag Berlin Heidelberg, German, (2009), 245.

[6]

L. E. Cárdenas-Barrón, A. A. Shaikh, S. Tiwari and G. Trevino-Garza, An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit, Comput. Indust. Engi., 139 (2020).

[7]

S. Chen, L. E. Cárdenas-barrón and J. Teng, Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity,, Inter. J. Prod. Eco., 155 (2014), 284-291.  doi: 10.1016/j.ijpe.2013.05.032.

[8]

K. Chung, S. Lin and H. M. Srivastava, The inventory models under conditional trade credit in a supply chain system, Appli. Mathem. Model., 37 (2013), 10036-10052.  doi: 10.1016/j.apm.2013.05.044.

[9]

C. Dye, C. Yang and F. Kung, The inventory models under conditional trade credit in a supply chain system, Appli. Mathem. Model., 37 (2013), 10036-10052.  doi: 10.1016/j.apm.2013.05.044.

[10]

P. M. Ghare and G. F. Schrader, A model for an exponential decaying inventory, J. Indust. Eng., 14 (1963), 238-243. 

[11]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, J. Opers. Res. Soci., 36 (1985), 335-338. 

[12]

J. Heizer, B. Render and C. Munson, Operations management: Sustainability and supply chain management, Pearson Education Inc, New Jersey, 2016.

[13]

Y. Huang, Economic order quantity under conditionally permissible delay in payment, European Journal of Operational Research, 176 (2007), 911-924.  doi: 10.1016/j.ejor.2005.08.017.

[14]

C. K. Jaggi, V. S. S. Yadavalli, A. Sharma and S. Tiwari, A fuzzy EOQ model with allowable shortage under different trade credit terms, Appli. Mathem.Inform. Sci., 10 (2016), 785-805. 

[15]

M. Khouja and A. Mehrez, Optimal inventory policy under different supplier credit policies, J. Manufact. Sys., 15 (1996), 334-339.  doi: 10.1016/0278-6125(96)84196-3.

[16]

R. Li, Y. Chan, C. Chang and L. E. Cárdenas-barrón, Pricing and lot-sizing policies for perishable products with advance-cash-credit-payments by a discounted cash-flow analysis, Inter. J. Prod. Eco., 193 (2017), 578-598. 

[17]

R. Li, K. Skouri, J. Teng and W. Yang, Seller's optimal replenishment policy and payment term among advance, cash, and credit payments, Inter. J. Prod. Eco., 197 (2018), 35-42. 

[18]

R. Li, J. Teng and Y. Zheng, Optimal credit term, order quantity and selling price for perishable products When demand depends on selling price, expiration date, and credit period, Annals Oper. Res., 280 (2019), 377-405.  doi: 10.1007/s10479-019-03310-2.

[19]

W. Luo and K. H. Shang, Technical note - managing inventory for firms with trade credit and deficit penalty, Oper. Res., (2019), 1–11.

[20]

P. Mahata, G. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, J. Sys. Sci. Oper. Logist., (2018), 1–17.

[21]

L. Ouyang, J. Teng, S. K. Goyal and C. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, Eur. J. Oper. Res., 194 (2009), 418-431.  doi: 10.1016/j.ejor.2007.12.018.

[22]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathem. Computer Model., 55 (2012), 367-377.  doi: 10.1016/j.mcm.2011.08.009.

[23]

N. H. Shah, Manufacturer-retailer inventory model for deteriorating items with price-sensitive credit-linked demand under two-level trade credit financing and profit sharing contract, Cogent Engin., 83 (2015), 1-14. 

[24]

N. H. Shah, Retailer's optimal policies for deteriorating items with a fixed lifetime under order-linked conditional trade credit, Uncert. Supp. Chain Manag., 5 (2017), 126-134. 

[25]

N. H. Shah, U. Chaudhari and M. Y. Jani, Optimal policies for time-varying deteriorating item with preservation technology under selling price and trade credit dependent quadratic demand in a supply chain, Inter. J. Appli. Comput. Mathem., 3 (2017), 363-379.  doi: 10.1007/s40819-016-0141-3.

[26]

H. Soni, N. H. Shah and C. Jaggi, Inventory models and trade credit?: A review, Contr. Cyber., 39 (2010), 867-882. 

[27]

L. Stemmler, The role of finance in supply chain management, Cost Management in Supply Chains, (2002), 165–176.

[28]

A. A. Taleizadeh, N. Pourmohammad-Zia and I. Konstantaras, Partial linked - to - order delayed payment and life time effects on decaying items ordering, Oper. Res., 2019.

[29]

A. A. Taleizadeh, D. W. Pentico, M. S. Jabalameli and M. Aryanezhad, An EOQ model with partial delayed payment and partial backordering, Omega, 41 (2013), 354-368.  doi: 10.1016/j.omega.2012.03.008.

[30]

J. Teng and K. Lou, Seller's optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits, J. Global Optim., 53 (2012), 417-430.  doi: 10.1007/s10898-011-9720-3.

[31]

P. Ting, Comments on the EOQ model for deteriorating items with conditional trade credit linked to order quantity in the supply chain management, Eur. J. Oper. Res., 246 (2015), 108-118.  doi: 10.1016/j.ejor.2015.04.046.

[32] J. Tirole, The theory of corporate finance, Princeton University Press, United State of America, 2010. 
[33]

S. Tiwari, L. E. Cárdenas-barrón, A. A. Shaikh and M. Goh, Retailer ' s optimal ordering policy for deteriorating items under order-size dependent trade credit and complete backlogging, Comput. Indust. Engin., 139 (2020).

[34]

Y. C. Tsao, Trade credit and replenishment decisions considering default risk, Comput. Indust. Engin., 117 (2018), 41-46.  doi: 10.1016/j.cie.2018.01.016.

[35]

Y. C. Tsao, R. P. F. R. Putri, C. Zhang and V. T. Linh, Opricing and ordering policies for perishable products under advance - cash - credit payment scheme, J. Indust. Engin. Inter., 15 (2019), 131-146. 

[36]

Va ndana and B. K. Sharma, An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages,, Mathem. Comput. Simul., 125 (2016), 99-112.  doi: 10.1016/j.matcom.2015.11.008.

[37]

W. Wang, J. Teng and K. R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, Eur. J. Oper. Res., 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027.

[38]

N. Wilson and B. Summers, Trade credit terms offered by small firms?: Survey evidence and empirical analysis, J. Busi. Fin. Account., 29 (2002), 317-351. 

[39]

J. Wu, F. Al-khateeb, J. Teng and L. E. Cárdenas-barrón, Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash- flow analysis, Inter. J. Prod. Eco., 171 (2016), 105-115. 

[40]

J. Wu, L. Ouyang, L. E. Cárdenas-barrón and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing,, Eur. J. Oper. Res., 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.

show all references

References:
[1]

A. A. A. Abuhommous, The impact of offering trade credit on firms' profitability, J. Cor. Account. Fina., 28 (2017), 29-40.  doi: 10.1002/jcaf.22298.

[2]

A. A. A. Aggarwal and S. P. Jaggi, Ordering policies of deteriorating items under permissible delay in payments, J. Opers. Res. Soci., 46 (1995), 658-662. 

[3]

D. Atnafu and A. Balda, He impact of inventory management practice on firms' competitiveness and organizational performance: Empirical evidence from micro and small enterprises in ethiopia, Cogent. Busi. Mag., 5 (2018).

[4]

Y. Benoist, P. Foulon and F. Labourie, Flots d'Anosov a distributions stable et instable differentiables, J. Amer. Math. Soc., 5 (1992), 33-74.  doi: 10.2307/2152750.

[5]

A. Cambini and L. Martein, Generalized convexity and optimization: Theory and application, Springer-Verlag Berlin Heidelberg, German, (2009), 245.

[6]

L. E. Cárdenas-Barrón, A. A. Shaikh, S. Tiwari and G. Trevino-Garza, An EOQ inventory model with nonlinear stock dependent holding cost, nonlinear stock dependent demand and trade credit, Comput. Indust. Engi., 139 (2020).

[7]

S. Chen, L. E. Cárdenas-barrón and J. Teng, Retailer's economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity,, Inter. J. Prod. Eco., 155 (2014), 284-291.  doi: 10.1016/j.ijpe.2013.05.032.

[8]

K. Chung, S. Lin and H. M. Srivastava, The inventory models under conditional trade credit in a supply chain system, Appli. Mathem. Model., 37 (2013), 10036-10052.  doi: 10.1016/j.apm.2013.05.044.

[9]

C. Dye, C. Yang and F. Kung, The inventory models under conditional trade credit in a supply chain system, Appli. Mathem. Model., 37 (2013), 10036-10052.  doi: 10.1016/j.apm.2013.05.044.

[10]

P. M. Ghare and G. F. Schrader, A model for an exponential decaying inventory, J. Indust. Eng., 14 (1963), 238-243. 

[11]

S. K. Goyal, Economic order quantity under conditions of permissible delay in payments, J. Opers. Res. Soci., 36 (1985), 335-338. 

[12]

J. Heizer, B. Render and C. Munson, Operations management: Sustainability and supply chain management, Pearson Education Inc, New Jersey, 2016.

[13]

Y. Huang, Economic order quantity under conditionally permissible delay in payment, European Journal of Operational Research, 176 (2007), 911-924.  doi: 10.1016/j.ejor.2005.08.017.

[14]

C. K. Jaggi, V. S. S. Yadavalli, A. Sharma and S. Tiwari, A fuzzy EOQ model with allowable shortage under different trade credit terms, Appli. Mathem.Inform. Sci., 10 (2016), 785-805. 

[15]

M. Khouja and A. Mehrez, Optimal inventory policy under different supplier credit policies, J. Manufact. Sys., 15 (1996), 334-339.  doi: 10.1016/0278-6125(96)84196-3.

[16]

R. Li, Y. Chan, C. Chang and L. E. Cárdenas-barrón, Pricing and lot-sizing policies for perishable products with advance-cash-credit-payments by a discounted cash-flow analysis, Inter. J. Prod. Eco., 193 (2017), 578-598. 

[17]

R. Li, K. Skouri, J. Teng and W. Yang, Seller's optimal replenishment policy and payment term among advance, cash, and credit payments, Inter. J. Prod. Eco., 197 (2018), 35-42. 

[18]

R. Li, J. Teng and Y. Zheng, Optimal credit term, order quantity and selling price for perishable products When demand depends on selling price, expiration date, and credit period, Annals Oper. Res., 280 (2019), 377-405.  doi: 10.1007/s10479-019-03310-2.

[19]

W. Luo and K. H. Shang, Technical note - managing inventory for firms with trade credit and deficit penalty, Oper. Res., (2019), 1–11.

[20]

P. Mahata, G. C. Mahata and S. K. De, An economic order quantity model under two-level partial trade credit for time varying deteriorating items, J. Sys. Sci. Oper. Logist., (2018), 1–17.

[21]

L. Ouyang, J. Teng, S. K. Goyal and C. Yang, An economic order quantity model for deteriorating items with partially permissible delay in payments linked to order quantity, Eur. J. Oper. Res., 194 (2009), 418-431.  doi: 10.1016/j.ejor.2007.12.018.

[22]

B. Sarkar, An EOQ model with delay in payments and time varying deterioration rate, Mathem. Computer Model., 55 (2012), 367-377.  doi: 10.1016/j.mcm.2011.08.009.

[23]

N. H. Shah, Manufacturer-retailer inventory model for deteriorating items with price-sensitive credit-linked demand under two-level trade credit financing and profit sharing contract, Cogent Engin., 83 (2015), 1-14. 

[24]

N. H. Shah, Retailer's optimal policies for deteriorating items with a fixed lifetime under order-linked conditional trade credit, Uncert. Supp. Chain Manag., 5 (2017), 126-134. 

[25]

N. H. Shah, U. Chaudhari and M. Y. Jani, Optimal policies for time-varying deteriorating item with preservation technology under selling price and trade credit dependent quadratic demand in a supply chain, Inter. J. Appli. Comput. Mathem., 3 (2017), 363-379.  doi: 10.1007/s40819-016-0141-3.

[26]

H. Soni, N. H. Shah and C. Jaggi, Inventory models and trade credit?: A review, Contr. Cyber., 39 (2010), 867-882. 

[27]

L. Stemmler, The role of finance in supply chain management, Cost Management in Supply Chains, (2002), 165–176.

[28]

A. A. Taleizadeh, N. Pourmohammad-Zia and I. Konstantaras, Partial linked - to - order delayed payment and life time effects on decaying items ordering, Oper. Res., 2019.

[29]

A. A. Taleizadeh, D. W. Pentico, M. S. Jabalameli and M. Aryanezhad, An EOQ model with partial delayed payment and partial backordering, Omega, 41 (2013), 354-368.  doi: 10.1016/j.omega.2012.03.008.

[30]

J. Teng and K. Lou, Seller's optimal credit period and replenishment time in a supply chain with up-stream and down-stream trade credits, J. Global Optim., 53 (2012), 417-430.  doi: 10.1007/s10898-011-9720-3.

[31]

P. Ting, Comments on the EOQ model for deteriorating items with conditional trade credit linked to order quantity in the supply chain management, Eur. J. Oper. Res., 246 (2015), 108-118.  doi: 10.1016/j.ejor.2015.04.046.

[32] J. Tirole, The theory of corporate finance, Princeton University Press, United State of America, 2010. 
[33]

S. Tiwari, L. E. Cárdenas-barrón, A. A. Shaikh and M. Goh, Retailer ' s optimal ordering policy for deteriorating items under order-size dependent trade credit and complete backlogging, Comput. Indust. Engin., 139 (2020).

[34]

Y. C. Tsao, Trade credit and replenishment decisions considering default risk, Comput. Indust. Engin., 117 (2018), 41-46.  doi: 10.1016/j.cie.2018.01.016.

[35]

Y. C. Tsao, R. P. F. R. Putri, C. Zhang and V. T. Linh, Opricing and ordering policies for perishable products under advance - cash - credit payment scheme, J. Indust. Engin. Inter., 15 (2019), 131-146. 

[36]

Va ndana and B. K. Sharma, An EOQ model for retailers partial permissible delay in payment linked to order quantity with shortages,, Mathem. Comput. Simul., 125 (2016), 99-112.  doi: 10.1016/j.matcom.2015.11.008.

[37]

W. Wang, J. Teng and K. R. Lou, Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime, Eur. J. Oper. Res., 232 (2014), 315-321.  doi: 10.1016/j.ejor.2013.06.027.

[38]

N. Wilson and B. Summers, Trade credit terms offered by small firms?: Survey evidence and empirical analysis, J. Busi. Fin. Account., 29 (2002), 317-351. 

[39]

J. Wu, F. Al-khateeb, J. Teng and L. E. Cárdenas-barrón, Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash- flow analysis, Inter. J. Prod. Eco., 171 (2016), 105-115. 

[40]

J. Wu, L. Ouyang, L. E. Cárdenas-barrón and S. K. Goyal, Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing,, Eur. J. Oper. Res., 237 (2014), 898-908.  doi: 10.1016/j.ejor.2014.03.009.

Figure 1.  Conceptual model for deteriorating products under order-linked trade credit
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Figure 2.  Inventory level
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Figure 3.  Graphical representation of Sub-case 1.1
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Figure 4.  Graphical representation of Sub-case 1.2
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Figure 5.  Graphical representation of Sub-case 2.1
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Figure 6.  Graphical representation of Sub-case 2.2
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Figure 7.  The graph of $ PTP^{}_{1.1}(p, M^{}, T^{}) $
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Figure 8.  The graph of $ PTP^{}_{1.1}(p, M^{}, T^{}) $
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Figure 9.  The graph of $ PTP^{}_{1.1}(p, M^{}, T^{}) $
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Table 1.  Summary of literature review
Author Payment Terms Time value of money Pers-pective Decision Variables Demand Function Deterio-ration Credit-risk customer
Sarkar [22] FTC No Buyer Cycle Time Time Time-varying No
Taleizadeh[29] PTC No Buyer Cycle Time Rate No No
Huang [13] Order Linked of FTC and PTC No Buyer Cycle Time Rate No No
Chen [7] Order Linked of FTC and PTC No Buyer Cycle Time Rate No No
Ting [31] Order Linked of FTC and PTC No Buyer Cycle Time Rate Constant No
Shah [24] Order Linked of FTC and PTC No Buyer Cycle Time; Selling Price Selling Price No No
Taleizadeh[28] Order Linked of FTC and PTC No Buyer Cycle Time Rate Time-varying No
Tiwari [33] Order Linked of FTC and PTC No Buyer Cycle Time Rate Constant No
Wang [37] FTC No Seller Cycle Time; Credit Period Credit Period Time-varying Yes
Shah [23] FTC No Seller Cycle Time; Credit Period Credit Period; Time No Yes
This Research Order Linked of FTC and PTC Yes Seller Cycle Time; Selling Price; Credit Period Selling Price; Credit Period Constant Yes
Note: FTC corresponds to full trade credit and PTC corresponds to partial trade credit
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Author Payment Terms Time value of money Pers-pective Decision Variables Demand Function Deterio-ration Credit-risk customer
Sarkar [22] FTC No Buyer Cycle Time Time Time-varying No
Taleizadeh[29] PTC No Buyer Cycle Time Rate No No
Huang [13] Order Linked of FTC and PTC No Buyer Cycle Time Rate No No
Chen [7] Order Linked of FTC and PTC No Buyer Cycle Time Rate No No
Ting [31] Order Linked of FTC and PTC No Buyer Cycle Time Rate Constant No
Shah [24] Order Linked of FTC and PTC No Buyer Cycle Time; Selling Price Selling Price No No
Taleizadeh[28] Order Linked of FTC and PTC No Buyer Cycle Time Rate Time-varying No
Tiwari [33] Order Linked of FTC and PTC No Buyer Cycle Time Rate Constant No
Wang [37] FTC No Seller Cycle Time; Credit Period Credit Period Time-varying Yes
Shah [23] FTC No Seller Cycle Time; Credit Period Credit Period; Time No Yes
This Research Order Linked of FTC and PTC Yes Seller Cycle Time; Selling Price; Credit Period Selling Price; Credit Period Constant Yes
Note: FTC corresponds to full trade credit and PTC corresponds to partial trade credit
Table 2.  Notations
Notation Description
$ i $ Index of case based on demand, $ i=\lbrack 1, 2\rbrack $
$ j $ Index of case based on time, $ j=\lbrack 1, 2\rbrack $
$ A $ Replenishment cost per order, in dollar
$ c $ Procurement cost per unit, in dollar
$ h $ The inventory holding cost rate per unit per unit time, in dollar
$ I_{e} $ Interest earned per dollar per unit time
$ I_{Loss} $ Interest revenue loss due to offering trade credit per dollar per unit time
$ r $ Annual compound interest rate per dollar per unit time
$ D(p, M) $ Annual demand rate per unit time, as a function of both $ p $and $ M $
$ F(M) $ The rate of default risk
$ \theta $ Deterioration rate, $ 0\le \theta \le 1 $
$ D_{d} $ The specific threshold in which permits the full trade credit
$ \alpha $ The fraction of the delay payments is permitted, $ 0\le \alpha \le 1 $
$ I(t) $ Inventory level at time $ t $
$ Q $ Seller's order quantity
$ OC $ The present value of ordering cost
$ HC $ The present value of holding cost
$ PC $ The present value of procurement cost
$ SR_{i} $ The present value of revenue in case i, $ i=\lbrack 1, 2\rbrack $
$ IE_{i, j} $ The present value of interest earned in case (i, j), $ i, j=\lbrack 1, 2\rbrack $
$ IL_{i, j} $ The present value of interest loss in case (i, j), $ i, j=\lbrack 1, 2\rbrack $
$ T $ Length of the replenishment cycle (decision variable)
$ M $ The credit period policies offered by the seller (decision variable)
$ p $ Selling price offered by the seller per unit (decision variable)
$ PTP(p, M, T) $ The present value of total annual profit, which is the function of $ p $, $ M $ and $ T $
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Notation Description
$ i $ Index of case based on demand, $ i=\lbrack 1, 2\rbrack $
$ j $ Index of case based on time, $ j=\lbrack 1, 2\rbrack $
$ A $ Replenishment cost per order, in dollar
$ c $ Procurement cost per unit, in dollar
$ h $ The inventory holding cost rate per unit per unit time, in dollar
$ I_{e} $ Interest earned per dollar per unit time
$ I_{Loss} $ Interest revenue loss due to offering trade credit per dollar per unit time
$ r $ Annual compound interest rate per dollar per unit time
$ D(p, M) $ Annual demand rate per unit time, as a function of both $ p $and $ M $
$ F(M) $ The rate of default risk
$ \theta $ Deterioration rate, $ 0\le \theta \le 1 $
$ D_{d} $ The specific threshold in which permits the full trade credit
$ \alpha $ The fraction of the delay payments is permitted, $ 0\le \alpha \le 1 $
$ I(t) $ Inventory level at time $ t $
$ Q $ Seller's order quantity
$ OC $ The present value of ordering cost
$ HC $ The present value of holding cost
$ PC $ The present value of procurement cost
$ SR_{i} $ The present value of revenue in case i, $ i=\lbrack 1, 2\rbrack $
$ IE_{i, j} $ The present value of interest earned in case (i, j), $ i, j=\lbrack 1, 2\rbrack $
$ IL_{i, j} $ The present value of interest loss in case (i, j), $ i, j=\lbrack 1, 2\rbrack $
$ T $ Length of the replenishment cycle (decision variable)
$ M $ The credit period policies offered by the seller (decision variable)
$ p $ Selling price offered by the seller per unit (decision variable)
$ PTP(p, M, T) $ The present value of total annual profit, which is the function of $ p $, $ M $ and $ T $
Table 3.  Interest earned and interest revenue loss cases
Case (1):$ D_{d}<D(p, M) $ Case (2):$ D_{d}\ge D(p, M) $
Sub-case1.1: $ 0\le M\le T\le T+M $ Sub-case 2.1: $ 0\le M\le T\le T+M $
Sub-case1.2: $ 0\le T\le M\le T+M $ Sub-case 2.2: $ 0\le T\le M\le T+M $
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Case (1):$ D_{d}<D(p, M) $ Case (2):$ D_{d}\ge D(p, M) $
Sub-case1.1: $ 0\le M\le T\le T+M $ Sub-case 2.1: $ 0\le M\le T\le T+M $
Sub-case1.2: $ 0\le T\le M\le T+M $ Sub-case 2.2: $ 0\le T\le M\le T+M $
Table 4.  A comparison of four cases:
Cases $ p^{} $(＄) $ M^{} $ (years) $ T^{} $ (years) Demand (items) $ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $ (＄)
Ordered-link trade credit 25.3312 0.081921 0.27852 375 5482.41
No ordered-link trade credit 26.2325 0.25352 344 5398.24
Full trade credit 25.505 0.054 0.27741 368 5478.92
Partial trade credit 25.7267 0.151 0.2979 372 5485.39
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Cases $ p^{} $(＄) $ M^{} $ (years) $ T^{} $ (years) Demand (items) $ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $ (＄)
Ordered-link trade credit 25.3312 0.081921 0.27852 375 5482.41
No ordered-link trade credit 26.2325 0.25352 344 5398.24
Full trade credit 25.505 0.054 0.27741 368 5478.92
Partial trade credit 25.7267 0.151 0.2979 372 5485.39
Table 5.  Summary of sensitivity analysis
Parameter $ p^{} $ $ M^{} $ $ T^{} $ $ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $
D$ _{d} $=350 25.3312 0.0819213 0.266878 5482.41
D$ _{d} $=450 25.5224 0.149572 0.297398 5493.47
c=6 23.7634 0.260868 0.360191 7123.23
c=8 24.6262 0.196586 0.320289 6280.3
c=10 25.5224 0.149572 0.297398 5493.47
c=12 26.4370 0.112410 0.28441 4761.02
c=14 27.3616 0.081181 0.277903 4081.91
h=0.6 25.5873 0.177748 0.360788 5518.11
h=0.8 25.5501 0.161851 0.324659 5505.19
h=1 25.5224 0.149572 0.297398 5493.47
h=1.2 25.5012 0.139723 0.275927 5482.67
h=1.4 25.4845 0.131596 0.258471 5472.62
$ \theta $=0.03 25.5515 0.162354 0.325811 5505.41
$ \theta $=0.04 25.5359 0.155573 0.310662 5499.3
$ \theta $=0.05 25.5224 0.149572 0.297398 5493.47
$ \theta $=0.06 25.5106 0.144213 0.285662 5487.86
$ \theta $=0.07 25.5002 0.139388 0.275185 5482.47
b=15 39.6903 0.291871 0.409115 11962.1
b=20 30.7678 0.198115 0.322033 7882.19
b=25 25.5224 0.149572 0.297398 5493.47
b=30 22.0502 0.114364 0.291447 3947.53
b=35 19.5771 0.084373 0.29072 2882.45
l=0.06 27.3895 0.790391 0.771212 5638.77
l=0.08 26.0368 0.328971 0.397466 5537.01
l=0.1 25.5224 0.149572 0.297398 5493.47
l=0.12 25.2051 0.0372622 0.268559 5476.34
l=0.14 25.1003 6.15627$ \times $10$ ^{-27} $ 0.266878 5475.06
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Parameter $ p^{} $ $ M^{} $ $ T^{} $ $ PTP^{}_{i.j}(p^{}, M^{}, T^{}) $
D$ _{d} $=350 25.3312 0.0819213 0.266878 5482.41
D$ _{d} $=450 25.5224 0.149572 0.297398 5493.47
c=6 23.7634 0.260868 0.360191 7123.23
c=8 24.6262 0.196586 0.320289 6280.3
c=10 25.5224 0.149572 0.297398 5493.47
c=12 26.4370 0.112410 0.28441 4761.02
c=14 27.3616 0.081181 0.277903 4081.91
h=0.6 25.5873 0.177748 0.360788 5518.11
h=0.8 25.5501 0.161851 0.324659 5505.19
h=1 25.5224 0.149572 0.297398 5493.47
h=1.2 25.5012 0.139723 0.275927 5482.67
h=1.4 25.4845 0.131596 0.258471 5472.62
$ \theta $=0.03 25.5515 0.162354 0.325811 5505.41
$ \theta $=0.04 25.5359 0.155573 0.310662 5499.3
$ \theta $=0.05 25.5224 0.149572 0.297398 5493.47
$ \theta $=0.06 25.5106 0.144213 0.285662 5487.86
$ \theta $=0.07 25.5002 0.139388 0.275185 5482.47
b=15 39.6903 0.291871 0.409115 11962.1
b=20 30.7678 0.198115 0.322033 7882.19
b=25 25.5224 0.149572 0.297398 5493.47
b=30 22.0502 0.114364 0.291447 3947.53
b=35 19.5771 0.084373 0.29072 2882.45
l=0.06 27.3895 0.790391 0.771212 5638.77
l=0.08 26.0368 0.328971 0.397466 5537.01
l=0.1 25.5224 0.149572 0.297398 5493.47
l=0.12 25.2051 0.0372622 0.268559 5476.34
l=0.14 25.1003 6.15627$ \times $10$ ^{-27} $ 0.266878 5475.06
Table 6.  Summary of sensitivity analysis (cont')
Parameter $ p $ $ M $ $ T $ $ PTP^_{i.j}(p^, M^, T^) $
$ \alpha $=0.48 26.3120 0.427639 0.398781 5544.5
$ \alpha $=0.64 25.7899 0.243915 0.329357 5510.66
$ \alpha $=0.8 25.5224 0.149572 0.297398 5493.47
$ \alpha $=0.96 25.3619 0.092792 0.281188 5484.03
$ \alpha $=1 25.3312 0.081921 0.27852 5482.41
r=0.024 25.9844 0.307269 0.445592 5546.65
r=0.032 25.7022 0.211592 0.349793 5516.32
r=0.04 25.5224 0.149572 0.297398 5493.47
r=0.048 25.3915 0.103826 0.264839 5475.44
r=0.056 25.2871 0.067011 0.242921 5460.93
I$ _{Loss} $=0.036 25.7848 0.242724 0.314041 5504.49
I$ _{Loss} $=0.048 25.6232 0.185330 0.304028 5497.72
I$ _{Loss} $=0.06 25.5224 0.149572 0.297398 5493.47
I$ _{Loss} $=0.072 25.4539 0.125260 0.292728 5490.55
I$ _{Loss} $=0.084 25.4043 0.107692 0.289277 5488.43
A=12 25.4177 0.120489 0.232408 5523.65
A=16 25.4741 0.136219 0.26707 5507.64
A=20 25.5224 0.149572 0.297398 5493.47
A=24 25.5651 0.161264 0.324661 5480.61
A=28 25.6034 0.171715 0.349611 5468.74
n=60 25.1003 1.57412$ \times $10$ ^{-9} $ 0.266878 5475.06
n=80 25.1003 6.67337$ \times $10$ ^{-9} $ 0.266878 5475.06
n=100 25.5224 0.149572 0.297398 5493.47
n=120 26.6491 0.452890 0.489132 5575.1
n=140 29.2678 1.039800 101428 5783.31
I$ _{e} $=0.024 25.4698 0.131901 0.258682 5472.69
I$ _{e} $=0.032 25.4934 0.139910 0.276078 5482.71
I$ _{e} $=0.04 25.5224 0.149572 0.297398 5493.47
I$ _{e} $=0.048 25.5589 0.161542 0.324322 5505.12
I$ _{e} $=0.056 25.6065 0.176891 0.359702 5517.93
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Parameter $ p $ $ M $ $ T $ $ PTP^_{i.j}(p^, M^, T^) $
$ \alpha $=0.48 26.3120 0.427639 0.398781 5544.5
$ \alpha $=0.64 25.7899 0.243915 0.329357 5510.66
$ \alpha $=0.8 25.5224 0.149572 0.297398 5493.47
$ \alpha $=0.96 25.3619 0.092792 0.281188 5484.03
$ \alpha $=1 25.3312 0.081921 0.27852 5482.41
r=0.024 25.9844 0.307269 0.445592 5546.65
r=0.032 25.7022 0.211592 0.349793 5516.32
r=0.04 25.5224 0.149572 0.297398 5493.47
r=0.048 25.3915 0.103826 0.264839 5475.44
r=0.056 25.2871 0.067011 0.242921 5460.93
I$ _{Loss} $=0.036 25.7848 0.242724 0.314041 5504.49
I$ _{Loss} $=0.048 25.6232 0.185330 0.304028 5497.72
I$ _{Loss} $=0.06 25.5224 0.149572 0.297398 5493.47
I$ _{Loss} $=0.072 25.4539 0.125260 0.292728 5490.55
I$ _{Loss} $=0.084 25.4043 0.107692 0.289277 5488.43
A=12 25.4177 0.120489 0.232408 5523.65
A=16 25.4741 0.136219 0.26707 5507.64
A=20 25.5224 0.149572 0.297398 5493.47
A=24 25.5651 0.161264 0.324661 5480.61
A=28 25.6034 0.171715 0.349611 5468.74
n=60 25.1003 1.57412$ \times $10$ ^{-9} $ 0.266878 5475.06
n=80 25.1003 6.67337$ \times $10$ ^{-9} $ 0.266878 5475.06
n=100 25.5224 0.149572 0.297398 5493.47
n=120 26.6491 0.452890 0.489132 5575.1
n=140 29.2678 1.039800 101428 5783.31
I$ _{e} $=0.024 25.4698 0.131901 0.258682 5472.69
I$ _{e} $=0.032 25.4934 0.139910 0.276078 5482.71
I$ _{e} $=0.04 25.5224 0.149572 0.297398 5493.47
I$ _{e} $=0.048 25.5589 0.161542 0.324322 5505.12
I$ _{e} $=0.056 25.6065 0.176891 0.359702 5517.93
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